Picturing Problems: Solving Logic Puzzles Diagrammatically John Howse, Gem Stapleton, Jim Burton, and Andrew Blake Centre for Secure, Intelligent and Usable Systems, University of Brighton, UK fJohn.Howse, g.e.stapleton, j.burton, [email protected] Abstract. Solving logic puzzles is a popular recreational activity. The solution of a logic puzzle involves understanding and reasoning about the information provided. Diagrammatic logics have been shown to help people to understand and reason about logical information. In this pa- per we use spider diagrams, a visual logic based on Euler diagrams, to visualize logic problems. Furthermore, we reason with the diagrammatic representation both syntactically and semantically until we reach a so- lution. We present four example logic puzzles of varying difficulty and produce their solutions in detail using spider diagrams. We suggest that the use of diagrams is helpful in their solution. 1 Introduction Logic puzzles have a long history. Lewis Carroll was an eminent protagonist, see [4], for example, with logical game-playing also occurring in Alice in Won- derland and Through the Looking Glass. Logic puzzles are usually expressed in the form of a list of textual statements from which some information can be deduced. The list of statements is frequently called the premise (or premises) and the information that can be deduced is called the conclusion. The solution involves logical reasoning to show that the conclusion follows from the premise. Frequently in logical puzzles, the conclusion is not stated explicitly, but given in the form of a question (or questions). The solver has to answer the question by reasoning about the premise. The simplest of these puzzles are of the form of classical syllogisms. For example, consider the following statements: 1. All Tigers are Cats 2. All Cats are Mammals 3. No Cats are Dogs Are all Tigers Mammals? Are all Tigers Dogs? As all Tigers are Cats and all Cats are Mammals, we can deduce that all Tigers are Mammals. Similarly, as all Tigers are Cats and no Cats are Dogs, we can deduce that no Tigers are Dogs. We can visualise this puzzle using Euler diagrams [5]. In figure 1, the first two statements of the puzzle are presented in an Euler diagram. From this diagram, Y.Sato and Z.Shams (Eds.), SetVR 2018, pp. 12-27, 2018. Picturing Problems: Solving Logic Puzzles... Howse, Stapleton, Burton and Blake we can observe that the circle labelled Tiger is within the circle labelled Mammal and hence that all Tigers are Mammals. In figure 2, the first and third statements of the puzzle are presented in an Euler diagram. From this diagram, we can observe that the circle labelled Tiger is disjoint from the circle labelled Dog and hence that no Tigers are Dogs. Fig. 1. Statements 1, 2 Fig. 2. Statements 1, 3 Fig. 3. Statements 1, 2, 3 In figure 3, all three statements are visualized. From this diagram, we can observe that all Tigers are Mammals and that no Tigers are Dogs. The circles labelled Mammal and Dog intersect because we are not given any information about the relationship between mammals and dogs; some dogs could be mammals and some dogs could not be mammals. The diagrams we are using do not have existential import; some of the regions in the diagram could represent the empty set. Hence, we cannot deduce that all dogs are mammals, even though we know from biological taxonomy that dogs are mammals. This is a lesson we can learn when considering logical puzzles and indeed logical statements generally: we should not make assumptions. We can only use the information provided { and, of course, anything we can deduce from that information. There is a widespread belief that the use of diagrammatic logics, such as Euler diagrams, is helpful in allowing people to understand and reason about logical information. One reason often cited for this is that some diagrammatic notations are well-matched. A notation is well-matched to meaning if its seman- tic relationships are matched by its syntactic relationships [7]. For example, in figure 1, the semantic relationship `All Tigers are Cats' (statement 1) is matched by the curve labelled Cat containing the curve Tiger in the diagrammatic rep- resentation. Similarly, in figure 2, the semantic relationship `No Cats are Dogs' (statement 3) is matched by the curves labelled Cat and Dog being disjoint in the diagrammatic representation. Diagram well-matchedness can help people to understand information. Other properties of diagrams can help people to reason about information. Diagrams can also contain free rides when compared to the same information represented textually (or symbolically) [11]. Free rides occur when the topological properties of a diagram force some logical consequence to be made explicit, where that consequence would otherwise take some inferential steps. For example, as discussed above, from statement 1, all Tigers are Cats, and statement 2, all Cats are Mammals, we can deduce that all Tigers are Mammals. However, in 13 Picturing Problems: Solving Logic Puzzles... Howse, Stapleton, Burton and Blake figure 1, we can observe directly that all Tigers are Mammals. As such, `all Tigers are Mammals' is an example of a free ride exhibited by the diagram in comparison to the textual representation. A similar free-ride, `No Tigers are Dogs', can be observed in figure 2 compared to the textual representation, where this information has to be deduced. The notion of free rides has been formalized and extended to the theory of observational advantages [13]. Thus free rides and observational advantages within diagrams can help people to reason about information. These helpful properties of some diagrams have been known for a long time: the notions of well-matchedness and free-rides are closely related to Peirce's concept of iconicity [10]. Many diagrammatic logics have been developed over the centuries that rep- resent sets and their relationships. Examples include Euler diagrams, Venn dia- grams [14] and Peirce's existential graphs [10]. Venn diagrams are often confused with Euler diagrams. However, in a Venn diagram all the intersections between sets are represented; shading is used to represent the empty set. Spider dia- grams [6] are based on Euler diagrams with the addition of syntax to represent individuals. Spider diagrams also use shading, borrowed from Venn diagrams, but to represent an upper bound on the cardinality of sets rather than just emptiness. They have well-defined reasoning rules [8] and are a fully formalized diagrammatic logic [9]. The iconicity and semiotics of spider diagrams are dis- cussed at length in [3]. We will use spider diagrams as the vehicle to visualize and solve logical puzzles. The mathematician Raymond Smullyan was a very prominent exponent of the art and science of recreational logic puzzle production. He is well-known for puzzles involving knights and knaves; knights always tell the truth and knaves always lie. Some of the puzzles considered in this paper are based on knights and knaves. In particular, they are adapted from Smullyan's The G¨odellian Puzzle Book [12], in which he uses puzzles to describe and prove G¨odel'sincompleteness theorem. Smullyan died in 2017 at the age of 97. This paper is dedicated to his memory. It begins in the strange land of knights and knaves. 2 Smullyan Island Smullyan Island is a very peculiar place. All the inhabitants are either Knights or Knaves but not both. Knights always tell the truth but Knaves always lie. The inhabitants of Smullyan Island are a playful lot and like to pose puzzles. However, you can't always believe what they say { unless they are Knights, of course. Unfortunately, you can't tell just by looking who is a Knight and who is a Knave. The first puzzle involves inhabitants of Smullyan Island. It is adapted from [12]. 2.1 The Prize Puzzle Three inhabitants of Smullyan Island are named Alice, Bob and Carol. At least one of them is a Knight and at least one is a Knave. Exactly one of them has a prize. They made the following statements: 14 Picturing Problems: Solving Logic Puzzles... Howse, Stapleton, Burton and Blake { Alice: Bob doesn't have the prize { Bob: I don't have the prize { Carol: I have the prize Who has the prize? Solution We start the solution by rephrasing the puzzle's context in diagrammatic form: The diagram, d1, states that there are exactly three individuals in the universe that we are considering: Alice, Bob and Carol, which we abbreviate to a, b and c, respectively. The shading tells us that there are no other elements except those represented by the dots. Diagram, d2, states that there is at least one Knight and at least one Knave. The shading indicates that all individuals are Knights or Knaves. The diagram, d3, states that there is exactly one Prize holder. These statements are invariants: they are assumed to be true. Having set up the puzzle's context, we can now consider the inhabitants' statements. For each of the inhabitants there are two cases. Either they are a Knight and therefore tell the truth or they are a Knave and therefore lie. We start by considering Alice's statement in the two cases. Alice is a Knight and therefore tells the truth and so Bob doesn't have the prize: Alice is a Knave and therefore is lying and so Bob does have the prize: We now consider Bob's statement in the two cases. 15 Picturing Problems: Solving Logic Puzzles... Howse, Stapleton, Burton and Blake Bob is a Knight and therefore tells the truth and so he doesn't have the prize: Bob is a Knave and therefore is lying and so he does have the prize: There are four possible combinations of Alice's and Bob's statements.